cp's OEIS Frontend

A359358 Let y be the integer partition with Heinz number n. Then a(n) is the size of the Young diagram of y after removing a rectangle of the same length as y and width equal to the smallest part of y.

%I A359358 #8 Dec 28 2022 09:05:02
%S A359358 0,0,0,0,0,1,0,0,0,2,0,1,0,3,1,0,0,2,0,2,2,4,0,1,0,5,0,3,0,3,0,0,3,6,
%T A359358 1,2,0,7,4,2,0,4,0,4,1,8,0,1,0,4,5,5,0,3,2,3,6,9,0,3,0,10,2,0,3,5,0,6,
%U A359358 7,5,0,2,0,11,2,7,1,6,0,2,0,12,0,4,4,13
%N A359358 Let y be the integer partition with Heinz number n. Then a(n) is the size of the Young diagram of y after removing a rectangle of the same length as y and width equal to the smallest part of y.
%C A359358 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%F A359358 a(n) = A056239(n) - A001222(n) * A055396(n).
%F A359358 a(n) = A056239(n) - A359360(n).
%e A359358 The partition with Heinz number 7865 is (6,5,5,3), which has the following diagram. The 3 X 4 rectangle is shown in dots.
%e A359358   . . . o o o
%e A359358   . . . o o
%e A359358   . . . o o
%e A359358   . . .
%e A359358 The size of the complement is 7, so a(7865) = 7.
%t A359358 Table[If[n==1,0,With[{q=Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Total[q]-q[[1]]*Length[q]]],{n,100}]
%Y A359358 The opposite version is A326844.
%Y A359358 Row sums of A356958 are a(n) + A001222(n) - 1, Heinz numbers A246277.
%Y A359358 A055396 gives minimum prime index, maximum A061395.
%Y A359358 A112798 list prime indices, sum A056239.
%Y A359358 A243055 subtracts the least prime index from the greatest.
%Y A359358 A326846 = size of the smallest rectangle containing the prime indices of n.
%Y A359358 A358195 gives Heinz numbers of rows of A358172, even bisection A241916.
%Y A359358 Cf. A124010, A243503, A268192, A316413, A325351, A325352, A326836, A326837, A355534, A359360.
%K A359358 nonn
%O A359358 1,10
%A A359358 _Gus Wiseman_, Dec 27 2022